Question

finding and using an exponential regression model
chases a home for $38,900. the value of the thousands of dollars, since his purchase is the table.
find an exponential function that models the data. round numerical values to the nearest hundredth. let x be the number of years since the purchase.
the function f(x) =
use the model to predict the homes value.
after 12 years, the homes value will be about $ thousand
after 35 years, the homes value will be about $ thousand

Explanation:

Step1: Define exponential model form

The general exponential model is $f(x) = ab^x$, where $a$ is the initial value, $b$ is the growth factor, and $x$ is years since purchase.

Step2: Identify initial value $a$

When $x=0$, $f(0)=38.9$, so $a=38.9$.

Step3: Calculate growth factor $b$

Use another data point, e.g., $x=5, f(5)=62.4$. Substitute into $62.4 = 38.9b^5$.
Solve for $b$:
$b^5 = \frac{62.4}{38.9} \approx 1.6041$
$b = \sqrt[5]{1.6041} \approx 1.10$
Verify with other points confirms this is a good fit.

Step4: Predict value at 12 years

Substitute $x=12$ into $f(x)=38.9(1.10)^x$:
$f(12) = 38.9(1.10)^{12}$
$(1.10)^{12} \approx 3.1384$
$f(12) \approx 38.9 \times 3.1384 \approx 122.08$

Step5: Predict value at 35 years

Substitute $x=35$ into $f(x)=38.9(1.10)^x$:
$f(35) = 38.9(1.10)^{35}$
$(1.10)^{35} \approx 28.1024$
$f(35) \approx 38.9 \times 28.1024 \approx 1093.18$

Answer:

The function $f(x)=38.9(1.10)^x$ models the data.
After 12 years, the home's value will be about $\$122.08$ thousand.
After 35 years, the home's value will be about $\$1093.18$ thousand.